Flexible structures having a large number of flexible modes can benefit from adaptive control techniques. These techniques are well suited to applications that have unknown modeling parameters and poorly known operating conditions. However, creating an accurate model of the dynamic characteristics of a structure can be extremely difficult if not impossible.
Most systems requiring closed-loop control have nonlinearities and uncertainties in their system dynamics. Computer simulation models of dynamical systems are very expensive to produce and cannot accurately represent important nonlinear system dynamics. Modern and classical control system design relies on linear plant models, leading to likely problems with unmodeled dynamics that can drive a system to instability. For many applications, it is a lengthy process to create a control system that is robust to unmodeled dynamics while achieving the desired performance. Often, it is only through encountering off-nominal situations that important nonlinear dynamics are understood. Unfortunately, this discovery may come too late, after a system is in use, leading to issues with safety, product integrity, performance, reliability, etc.
Various methods are known in the art for dealing with nonlinear control problems. For example, use of fixed gain controllers with gain scheduling is known for use with aircraft flight control systems, where gains are scheduled for different expected operating conditions around which a linear model is created. Notch filters are known for use with frequencies that can interact with the control system or excite resonant frequencies. For example, flight control systems may use notch filters to prevent commands issued from the control system to excite aircraft wing flexible modes that could lead to destructive behavior, for example, due to resonance. All physical systems experience external disturbances that have varying effects on system behavior. These disturbances can have potentially profound effects on systems, especially when the disturbances are persistent (i.e., continually recurring or slowly decaying). Disturbance rejection is known, where the disturbance must be modeled accurately so that it can be accommodated or rejected from the plant output states and so that it does not enter the feedback loop. However, it is not always possible to know and accurately model the disturbance in advance.
A physical system or “plant” can often be usefully modeled as a linear, time-invariant, finite dimensional system:
                    {                                                                                                  x                    .                                    p                                =                                                      Ax                    p                                    +                                      Bu                    p                                    +                                      Γ                    ⁢                                                                                  ⁢                                          u                      D                                                                                                                                                                                    y                    p                                    =                                      Cx                    p                                                  ;                                                                            x                      p                                        ⁡                                          (                      0                      )                                                        =                                      x                    0                                                                                      }                            (        1        )            where the plant state xp(t) is an Np-dimensional vector, the control input vector up(t) is M-dimensional, and the plant output vector yv(t) is P-dimensional. A, B, and C are constant matrices of the appropriate dimensions, and the notation (A, B, C) is commonly used to denote a plant modeled by (1). r is a constant matrix related to the disturbance vector, uD(t), which is MD-dimensional and can be thought to come from a disturbance generator:
                    {                                                                              u                  D                                =                                  Θ                  ⁢                                                                          ⁢                                      z                    D                                                                                                                                                                                      z                      .                                        D                                    =                                      F                    ⁢                                                                                  ⁢                                          z                      D                                                                      ;                                                                            z                      D                                        ⁡                                          (                      0                      )                                                        =                                      z                    0                                                                                      }                            (        2        )            where the disturbance state zD(t) is ND-dimensional. (All matrices in (1) and (2) have the appropriate compatible dimensions.) Eqs. (1) and (2) have been used to describe systems having persistent disturbances having known form but unknown amplitude. The disturbance generator can also be rewritten in a form that is not a dynamical system, but is sometimes easier to use:
                    {                                                                              u                  D                                =                                  Θ                  ⁢                                                                          ⁢                                      z                    D                                                                                                                                            z                  D                                =                                  L                  ⁢                                                                          ⁢                                      ϕ                    D                                                                                      }                            (        3        )            where θD is a vector composed of the known basis functions for the solutions of uD=zD, i.e., θD consists of the basis functions which make up the known form of the disturbance, and L is a matrix of dimension ND by dim(θD).
In much of the control literature, it is assumed that the plant and disturbance generator parameter matrices A, B, C, Γ, Θ, F are known. This knowledge of the plant and its disturbance generator allows the separation principle of linear control theory to be invoked to arrive at a state-estimator-based linear controller that can stabilize the plant and suppress the persistent disturbances via feedback. However, in many systems, the plant is poorly known; only the form(s) of the disturbances are known but not the amplitudes, and other methods are required.
A practical and well-accepted representation of flexible structures is based on the finite element method (FEM). The FEM of the lumped model in physical coordinates q, for a linearized actively controlled flexible structure with M control inputs, and P plant outputs is given in matrix form as
                    {                                                                                                                        M                      0                                        ⁢                                          q                      ~                                                        +                                                            D                      0                                        ⁢                                          q                      .                                                        +                                                            K                      0                                        ⁢                    q                                                  =                                                      B                    0                                    ⁢                  ι                                                                                                                          y                  p                                =                                                                            C                      0                                        ⁢                    q                                    +                                                            E                      0                                        ⁢                                          q                      .                                                                                                                              (        4        )            
This system can be put into a modal form with the transformationq=Φ0η  (5)where
                    {                                                            I                ≡                                                      Φ                    0                    T                                    ⁢                                      M                    0                                    ⁢                                      Φ                    0                                                                                                                                            Λ                  0                                ≡                                                      Φ                    0                    T                                    ⁢                                      K                    0                                    ⁢                                      Φ                    0                                                  ≡                                  diag                  ⁡                                      [                                          ω                      k                      2                                        ]                                                                                      }                            (        6        )            (I is the identity matrix of appropriate dimension.)
Using the transformation (5), the modal form of (4) follows:
                    {                                                                                                  η                    _                                    +                                                                                    D                        _                                            0                                        ⁢                                          η                      .                                                        +                                                                                    Λ                        _                                            0                                        ⁢                    η                                                  =                                                                            B                      _                                        0                                    ⁢                  u                                                                                                                          y                  p                                =                                                                                                    C                        _                                            0                                        ⁢                    η                                    +                                                                                    E                        _                                            0                                        ⁢                                          η                      .                                                                                                          }                            (        7        )            
This system can be put into a modal first-order form with the states
                              x          p                ≡                  [                                                    η                                                                                      η                  .                                                              ]                                    (        8        )            
Many kinds of systems have modal forms, and the results apply to control of any such system, not just flexible structures. The control of these physical systems or “plants” is straightforward if the plant satisfies the requirement of “Almost Strict Positive Realness,” wherein the matrix product CB is positive definite and the system transfer function from the plant output to the plant input is minimum phase (e.g., all zeros of the numerator are in the left half-plane).